In recent years many problems concerned with the dispersion of a passive contaminant along pipes and channels have been investigated, and this paper is concerned with one such problem which arises in diverse applications. This is the study of the longitudinal dispersion of a contaminant whose concentration is prescribed as a harmonic function of time at one cross-section. On the basis of physical arguments and of detailed calculations for two laminar flows it is shown that for high frequencies the concentration pattern is transported downstream at the maximum fluid velocity but that for low frequencies it is transported at the discharge velocity, and that the fluctuations in concentration decay to zero in a much shorter downstream distance for high frequencies than for low frequencies. It is shown further that at high frequencies the concentration is exponentially small except near the places where the fluid velocity attains its maximum, whereas for low frequencies the variation in concentration over the cross-section is small. Some of these conclusions are compared with those made by others, and the agreement is in general satisfactory.